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Classical probability preserves amplitude, quantum preserves |amplitude|^2.
They're different things, and they could, potentially, be even more different.
Um, but isn't that just a convention? Why should we treat the "amplitude" of a classical probability as being the probability?
Does the problem have something to do with the extra directionality quantum probabilities have by virtue of the amplitude being in C? (so that |0> and (-1*|0>) can cancel each other out)
Classical probability transformations preserve amplitude and quantum ones preserve |amplitude|^2. That's not a whole reason, but it's part of one.
Yes, that's part of the difference. Quantum transformations are linear in a two-dimensional wave amplitude but preserve a 1-dimensional |amplitude|^2. Classical transformations are linear in one-dimensional probability and preserve 1-dimensional probability.
Ah, I get it now, thanks!
(Copenhagen is still wrong though ;)